Titles and abstracts
Structures in Enumerative Geometry
The University of Sheffield, 23-27 January 2023
Luca Battistella: Alternative compactifications of the moduli space of curves
Francesca Carocci: Logarithmic and tropical techniques in moduli theory
Jeongseok Oh: Counting sheaves on Calabi-Yau 4-folds
Alessio Corti: Cluster varieties and degenerations of Fano varieties
I will present a conjecture, sketch a proof in dim = 2, and present some evidence
Michel van Garrel: The Prism of Intrinsic Mirror Symmetry
In mirror symmetry as for vinyl records, there are two sides, an A-side and a B-side. Unlike vinyl records though, both sides are supposed to be equivalent. This correspondence is usually proven through the computation of each side, which limits the scope of results. Intrinsic Mirror Symmetry by Gross and Siebert changes the game. The full enumerative invariants of the A-side construct the B-side. This is the mirror construction. Then the mirror theorem becomes a prism (period integrals) applied to the B-side in order to recover specific enumerative invariants of the A-side. In joint work with Ruddat and Siebert, we show how this works for log Calabi-Yau varieties with smooth boundary, such as Fanos with smooth anticanonical divisor.
Dominic Joyce: The structure of invariants counting coherent sheaves on complex surfaces
Let X be a complex projective surface with geometric genus pg > 0. We can form moduli spaces M(r,a,k)st ⊂ M(r,a,k)ss of Gieseker (semi)stable coherent sheaves on X with Chern character (r,a,k), where we take the rank r to be positive. In the case in which stable = semistable, there is a (reduced) perfect obstruction theory on M(r,a,k)ss, giving a virtual class [M(r,a,k)ss]virt in homology.
David Kern: Derived geometric CohFT for orbifold quasimap theory
The Gromov–Witten invariants of a smooth projective variety produce a Cohomological Field Theory, a certain algebraic structure controlled by the homologies of the moduli stacks of stable curves. Mann–Robalo showed that, using derived geometry, it can be lifted from the cohomological setting to the geometric one.
When the target is a stack, it is known from Abramovich–Graber–Vistoli that the CohFT is only exhibited on (a “cyclotomic” decomposition of) its inertia stack. I will explain how the orbifold structure of the target can be used to extend the GW stability condition to the family of quasimap theories on it, and how Mann–Robalo’s construction adapts to a geometric CohFT in which the cyclotomic inertia appears naturally.
Navid Nabijou: Roots and logs in the enumerative forest
Logarithmic and orbifold structures provide two different paths to the enumeration of curves with fixed tangencies to a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive. I will discuss joint work with Luca Battistella and Dhruv Ranganathan, in which we identify birational invariance as the key property distinguishing the two theories. The logarithmic theory is stable under strata blowups of the target, while the orbifold theory is not. By identifying a suitable system of blowups, we define a “limit" orbifold theory and prove that it coincides with the logarithmic theory. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors, where the situation is much-better understood. Time permitting, I will discuss related upcoming work (with the same coauthors), in which we apply techniques from toroidal intersection theory to the study of invariants with negative tangency orders.
Nicola Pagani: A wall-crossing formula for universal Brill-Noether classes
We will discuss an explicit graph formula, in terms of boundary strata classes, for the wall-crossing of universal (=over the moduli space of stable curves) Brill-Noether classes. More precisely, fix two stability conditions for universal compactified Jacobians that are on different sides of a wall in the stability space. Then we can compare the two universal Brill-Noether classes on the two compactified Jacobians by pulling one of them back along the (rational) identity map. The calculation involves constructing a resolution by means of subsequent blow-ups. If time permits, we will discuss the significance of our formula and potential applications. This is joint with Alex Abreu.
Renata Picciotto: The derived moduli of sections and virtual pushforwards
Derived algebraic geometry provides a powerful set of tools to enumerative geometers, giving geometric spaces which encode the "virtual structures" of the moduli problems . I will discuss a joint work with D. Kern, E. Mann and C. Manolache in which we define a derived enhancement for the moduli space of sections. This enriched space neatly encodes the perfect obstruction theory and virtual structure sheaves of many theories. Special cases include Gromov-Witten and quasimaps theories. To illustrate the potential of this approach, I will explain how we use local derived charts to prove a virtual pushforward formula between stable maps and quasimaps without relying on torus localization.
Dhruv Ranganathan: Logarithmic Gromov-Witten theory and the double ramification cycle
The logarithmic Gromov-Witten theory of the pair (X,D) where X is toric and D is the toric boundary is probably the most basic target geometry in the subject and, in retrospect, was hiding behind work of Mikhalkin and Nishinou-Siebert in the early 2000s on tropical correspondence. I will explain how these logarithmic GW invariants can be expressed as products of natural tautological classes and double ramification cycles in the “logarithmic” tautological ring of the moduli space of curves. Practically carrying out calculations leads naturally to tropical geometry and the correspondence theorems. The results rely on a pleasant mix of ingredients: logarithmic birational invariance, product formulas, strict transform formulas in intersection theory, and a curious gadget called the logarithmic algebraic torus. I’ll try to give a sense for how these ideas fit together, and the broader context for the results. This is joint work with Sam Molcho (ETH) and Ajith Urundolil Kumaran (Cambridge).
Richard Thomas: Counting sheaves by counting curves